A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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On the difference between a modification of a stochastic process and two indistinguishable processes.

I am following Brownian motion and Stochastic calculus by Karatzas and Shreve They give the following definitions with an example: I do not understand the difference between definition 1.1 and 1.3 ...
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8 views

Finding a predictable process $\lambda$ such that $dA_t = d\langle M \rangle_t \lambda_t$ where $Y = M + A$ is an exponential Lévy process

Assume $X$ is a finite-variation Lévy process and let $Y_t = e^{X_t}$. It can be shown that one can decompose $Y_t$ as $Y_t = M_t + A_t$ where $M$ is a local martingale and $A$ is continuous and ...
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29 views

Let $N$~Pois$(\lambda)$, $X|(N=n)$~Bin$(N,p)$, $Y=N-X$. Show $X$, $Y$ are independent and Poisson with parameters $\lambda p$ and $\lambda (1-p)$.

Any direction on this problem would be much appreciated. So far I know the joint distribution of $X$ and $Y$ is $\begin{align} \mathsf P(X=x, Y=y) & = \mathsf P(X=x, N-X=y) \\ & = \mathsf ...
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39 views

Integral with respect to Brownian motion, Variance

Good day. Imagene we have a martingale $M(t)=\int_0^t f(s)dB(s)$ which satisfies Dambis-Dubins-Schwarz Theorem. At the same time $M(t)^2 - <M>(t)$ is a Martingale starting in $0$ as well. If i ...
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19 views

Expected number of arrival in a poisson process which are not followed by arrival in next time $\delta$ .

Let $a,\lambda,\delta > 0$ . Compute the expected number of arrivals in a Poisson process with intensity function $\alpha(t)=ae^{-\lambda t}$, which are not followed by another arrival with time ...
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1answer
23 views

If $N(t)$ is a Poisson process with parameter $\lambda(t)$ then is $N'(t)=N(t+2)-N(2)$ a poisson process?

If $N(t)$ is a Poisson process with parameter $\lambda(t)$ then is $N'(t)=N(t+2)-N(2)$ a poisson process? I think it should be poisson process as it is like observing a poisson process after time $2$ ...
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1answer
54 views

Conditional distribution of mixed process

$$ N(t)=(1-B)\cdot N_0(t)+B\cdot N_1(t), \quad \quad \text{where $B$ is Bernoulli($p$), $N_0(t) \sim \operatorname{Poiss}(\lambda_0 t)$ and $N_1(t) \sim \operatorname{Poiss}(\lambda_1 t)$}. $$ I ...
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2answers
18 views

On the definition of continuous time martingales Stroock Varadhan $\times$ Kallenberg

In the definition of martingales, one finds in Stroock and Varadhan (Multidimensional Diffusion processes - page 20) the strange request that it be right-continuous process. However no such ...
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9 views

Death process - median time to die out

Question A pure death process {X(t); t>=0}, where X(t) denotes the number of individuals alive at time t, starts with X(0) = 8. The lifetime of each of these individuals is exponential with mean 1/υ. ...
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1answer
30 views

On the proof of lemma 1.2.4 of Stroock and Varadhan A question concerning stopping times

In the book Multidimensional diffusion processes, of Stroock and Varadhan one reads (page 23): This is the proof of $(i)$. Here the authors say Define $f_t$ on $(\{\tau \leq t\}, \mathcal{F}_t ...
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4answers
62 views

Supermartingale vanishing at some stopping time

Let $\left\{X_t\right\}_{t\in[0, T]}$ be a continuous and non-negative supermartingale. We define the stopping time $$\tau_0:=\inf\{t\in[0,T]:X(t)=0\}\wedge T$$ and immediately obtain by continuity ...
5
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2answers
48 views

Suppose $\xi_1, \xi_2,\ldots$ are i.i.d. random variables with mean $\mu$, variance $\sigma^2$. Form the random sum $S_{N} = \xi_{1}+\cdots+\xi_{N}$.

(a) Derive the mean and variance of $S_{N}$ when $N$ has Poisson distribution with parameter $\lambda$. So far, for the mean, I have the following: $E[S_{N}] = E[E[S_{N}\mid N=n]]$ $$ = ...
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1answer
16 views

Do all Stochastic matrix have a stationary probability vector?

I know that a stochastic matrix will have 1 as one of its eigenvalues. But do the stochastic matrices all have a stationary probability vector? Basically, could there be a case where the eigen ...
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38 views

Difference between the “Hazard Rate” and the “Killing Function” of a diffusion model?

I posted this question on Cross Validated - but I think it applies here too. Also, it increases the chances of the question being answered. Link here If this is not acceptable - administrators ...
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19 views

a conceptual question on markov chain [duplicate]

Suppose $\{X_n,n\ge 0\}$ and $\{Y_n,n\ge0\}$ are two independent discrete-time markov chains (DTMC) with state space $S=\{0,1,2,\ldots\}$. Prove or give a counterexample to: $\{X_n+Y_n,n\ge 0\}$ is ...
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1answer
28 views

A credit model. Default time.

In a paper, I find the following situation: Let $(\Omega,\mathcal{G},\mathbb{Q})$ be a probability space. $\mathbb{Q}$ is supposed to be a risk neutral measure. Suppose that ...
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0answers
23 views

How to obtain accumulated counts of past events by time $t$?

Given $f: [0, \infty) \to \{0,1\}$, $f(t)$ represents whether there is an event occurring at time $t$. How can we obtain $g: [0,\infty) \to \mathbb{N}_0$ so that $g(t)$ represents the number of ...
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31 views

Showing that $(X_n)$ obeys the Markov Property. [on hold]

Consider a process $(X_n)_{n\geq0}$ where we define $X_0 = 0$ and for $n \geq 1$: $$X_n = X_{n-1} + Z_n$$ where $Z_n$ for $n \geq 1$ are independent random variables on $\{ -1, 1 \}$ with ...
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19 views

Compute distribution in Hidden Markov models

Let $Z_1, Z_2, ..., Z_n$ be the latent variables, and $X_1, X_2, ... X_n$ be the observed ones in a hidden markov models. Let's assume that the parameters of the hidden Markov models are known: the ...
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0answers
15 views

Density of intersection of sets with boundary condition

I would like to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{n}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) : ...
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21 views

What is the interpretation of $\nu(dy - x)$ where $\nu$ is a Lévy measure?

In a paper I am reading, it is seemingly suggested that, if $\nu(dx)$ is a Lévy measure, then the following holds for a function $f(x)$ which is smooth (and satsifies some integrability conditions): ...
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1answer
37 views

hitting times and stopping times

stopping times are always hitting times, but not the other way around. As an example of this, Last exit times are not stopping times as they depend on future information. the last exit time of $A$: ...
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1answer
45 views

Age distribution when meeting

I have a question regarding Poisson process. I will tell the story in the context of a player-monster game. Consider a player who is born at $t=0$. He will win the game if he can survive until ...
2
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1answer
32 views

Probability that a stochastic process is below a special random level

Given a stochastic process $x(t)$ over time $t \in [0,T]$, and a given (deterministic) $\tau$, where $0<\tau<T$, define a random variable $x^{*}$ as $$ x^{*} \triangleq \inf\bigg\{y: ...
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1answer
21 views

Weak convergence for composition of cadlag stochastic processes

Let $(X^n_t)_{t \geq 0}$ be a sequence of cadlag stochastic processes, that is $X^n$ is a random element in the Skorokhod space $D([0, \infty), \mathbb{R}$) for each $n \in \mathbb{N}$. Also for each ...
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14 views

Lognormal approximation of the sum of successive values of a lognormal process

I would like to use a lognormal process to approximate the successive values of another lognromal process. Let $X_t$ be a lognormal process. I would like to approximate $$ Y_t := \sum_{t=0}^T X_t $$ ...
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1answer
20 views

Predictability of $\int^t_0 f(X_s)\,\mathrm ds$ where $X$ is a Lévy process

Let $X_t$ be a Lévy process and $f(x)$ some smooth function. Under what conditions is $$ Y_t = \int^t_0 f(X_s)\,\mathrm ds$$ predictable? Not sure how to investigate this. It is clearly adapted, so ...
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1answer
17 views

Find $P(\eta_t=m)$, $m=0,1,2,\dots,$

Let $\epsilon_t$, $t=1,2,\dots$ independent random variables with $P(\epsilon_t=1)=p$ and $P(\epsilon_t=-1)=1-p$. If $\eta_0=0,\eta_t=\eta_{t-1}+\epsilon_t$ , $t=1,2,\dots$ where $\eta_t$ is ...
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13 views

Stopping of quadratic covariaton

I am given two local martingales $M$ and $N$ and a stopping time $\tau$. We work on a finite time interval $[0,T]$. I want to prove $$\langle M,N\rangle^{\tau}=\langle M^\tau,N\rangle$$ using the ...
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1answer
17 views

Expected value of distance between independent Brownian motions

Suppose $\{W^{(1)}_t, t\geq 0\}$ and {$W^{(2)}_t, t\geq 0\}$ are two independent Brownian motions. If I recall correctly, the distance between the two at a given time has the following property: ...
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48 views

one dimensional SDE with zero drift

I was trying to prove that the solution $X$ to the one dimensional SDE $dX_t = \sigma(X_t)dW_t$ (where $\sigma$ is a real valued Borel measurable function, $W$ is a 1d Brownian Motion) cannot explode, ...
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1answer
37 views

Measurability of marginal distributions of a random measurable function

For a probability space $(\Omega, \mathcal F, \mathsf P)$, let $X \colon \Omega \times [0,1] \to \mathbf R \colon (\omega, t) \mapsto X(\omega,t)$ be a random Borel function (i.e. an $(\mathcal ...
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42 views

Algebra behind Feynman-Kac theorem?

According to many sources, The Feynman-Kac theorem in Equation (1) below is the solution to Equation (3) - if X(t) follows a diffusion such as in (2). (Most Important) - Can someone show the algebra ...
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1answer
25 views

Covariance of Ornstein-Uhlenbeck process

$U(t)=e^{-\mu t}W(\frac{\sigma^2e^{2\mu t}}{2\mu})$. The problem is to find $Cov[U(t),U(t+s)]$. I used the identity, $W(\frac{\sigma^2e^{2\mu t}}{2\mu})=W(\frac{\sigma^2e^{2\mu t}e^{2\mu s}}{2\mu ...
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16 views

Stochastic variation in stockexchange, weather sciences [on hold]

As the Weierstrass continuous function has no derivative defined, its curvature or differential equations of the function is meaningless. Is that correct? Is there a definition of sufficiently ...
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1answer
71 views

Does this game make you arbitrarily rich with probability one?

We toss a coin. If it's heads we win $\$ 1$, otherwise we lose $ \$ 1$. Fix some large sum. Will we be winning this amount with probability one at some point? We assume that we have infinitely many ...
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11 views

Is there a Little law for a network of two connected queues?

From Patterson et al' Computer Organization and Design: Throughput and Response Time Do the following changes to a computer system increase throughput, decrease response time, or both? ...
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17 views

Stochastic optimal control : infinite horizon problem

Assume an investor has utility function $U(C_t)=\frac{C_t^\gamma}{\gamma}$. The investor wishes to consume some of their wealth at a rate $C_t$ per unit time, and invest in both risk-free bonds and a ...
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1answer
9 views

Common factors in ARIMA(p,d,q)

I have some concerns regarding interpreting ARIMA processes, A general ARIMA process is on the form $$ \phi(B)X_t = \theta(B)Z_t,\,\,Z_t\sim WN(0,\sigma^2)$$ For example if I have $$Y_t = ...
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2answers
51 views

piecewise weak convergence in $C[0,1]$

Let $P$ and the sequence $P_n$ be probability measures on $C[0,1]$ with the uniform metric. Fix $0<u<1$ and let $\Pi_1$ and $\Pi_2$ be the projections from $C[0,1]$ onto $C[0,u]$ and $C[u,1]$, ...
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What are the statistics of the discrete Fourier transform of a Bernoilli process?

The problem I would like to understand the statistics of the discrete Fourier transform of a sequence of uncorrelated events $\{x_n\}$ each of which takes the value $\pm1$ with probability $1/2$. In ...
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1answer
37 views

Show that if $\{X_n\}$ is a Markov Chain

Show that, if $\{X_n\}$ is a Markov Chain then $$P(X_n=j\mid X_k=l,X_m=i)=P(X_n=j\mid X_m=i),0\leq k<m<n$$ What I did is $$P(X_n=j\mid ...
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19 views

What is the Skewness of a Geometric Brownian Motion?

Consider a GBM : $$S(t) = S(0)\exp\left({(\mu-\frac{1}{2}\sigma^2) t + \sigma W_t}\right)$$ $$d\log S(t) = (\mu-\frac{1}{2}\sigma^2) t + \sigma dW_t$$ $$\frac{d S(t)}{S(t)} = \mu t + \sigma ...
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1answer
79 views
+50

How can I solve this stochastic system of equation?

$(B_1(t),B_2(t))$ is a 2-dimensional standard Brownian motion. $\alpha , \beta$ are constant. The system of equations is: $$dX_1(t)=X_2(t)dt+\alpha dB_1(t)\\dX_2(t)=-X_1(t)dt+\alpha dB_2(t)$$ I tried ...
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9 views

Measure of Sample Paths that Never Cross LIL Bound

Suppose that $X_i$ is an i.i.d. sequence of random variables, with $P(X_i=1)=P(X_i=0)=1/2$. Then $S_n = \sum_{i=1}^n$ is a zero mean random walk. From the Law of the Iterated Logarithm, for all ...
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21 views

Process Transition Algorithm

I have a process with 100 possible states and independent entities going through the process. All the Entities have been observed through a span of 5 years at the end of each month. When the ...
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15 views

Are some some particular subspaces of cadlag functions Polish?

Consider the space $D := D((0, \infty), \mathbb{N})$ of cadlag functions $f : (0, \infty) \to \mathbb{N}$ equipped with the Skorokhod $M_1$-topology. Then $D$ is Polish. Question 1: I want to check ...
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25 views

The definition of Random time. [closed]

I define a random time of a Martingale $ \lbrace Z_n: n \geq 1 \rbrace $ to be the random variable $ N $ for which there is a function $ f(Z_1, Z_2,..., Z_n) $ such that $ P \lbrace N=n | ...
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13 views

Exact solution to nonlinear backward SDE

I have read a paper about numerical SDE. After deriving the method, it uses the method to calculate the following nonlinear cases: $$\begin{cases} dX_t=ud\tau+\sigma ...
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15 views

Probability Law of Stochastic Process Definition

I am reading Probability and Stochastics by Çınlar, and am confused by the following definition in it: I must be missing something because this definition does not seem correct to me. For ...